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It is the winter of 1961, and Wally has turned up at the Lovelace Clinic in Albuquerque, New Mexico, to begin a series of medical tests – 87 of them, to be exact. Wally will have to swallow three feet of rubber hose and drink a pint of radioactive water; later, Wally will head to Oklahoma City to spend a record ten and a half hours in an isolation tank, immersed in water and complete darkness and silence: an attempt to simulate the weightless darkness of space. The absence of stimuli can provoke distress and hallucinations; Wally stays cool and in control, although afterwards will admit to taking a few catnaps. Wally passes with flying colours.

These tests had been devised by Dr William Randolph Lovelace II to discover whether the subjects had what Tom Wolfe would call the right stuff. They were the very same tests undergone by Alan Shepard, Gus Grissom, Gordon Cooper, Wally Schirra, Deke Slayton, John Glenn and Scott Carpenter, known as the Mercury Seven, America’s very first astronauts. Shepard became the first American in space in May 1961; Glenn the first American in orbit less than a year later. But Wally never made it out of the Earth’s atmosphere. Because Wally is a girl.

And that’s what she calls herself – none of this “woman” stuff for her – to this very day. Mary Wallace Funk was born 79 years ago in New Mexico and had a pilot’s licence by her teens. At 20 she was a civilian flight instructor at Fort Sill, Oklahoma – putting the pilots of the United States Army through their paces. And at 21 she became the youngest of the group of women now known, belatedly, as the Mercury 13. This baker’s dozen was a remarkable crew – although unlike their male counterparts, who benefited from the camaraderie of the military and undertook Lovelace’s tough regime in groups, they had to go solo through the medical obstacle course.

Although the “Woman in Space” programme, as it was known, was never official, Nasa had wondered whether a woman’s size (small) and weight (light) might make her especially suitable for the cramped conditions of a space capsule. But the dreams of the Mercury 13 were extinguished by a series of 1962 Congressional hearings that ensured that women would not become part of the space programme for a long while: “The fact that women are not in this field is a fact of our social order,” testified John Glenn. In 1963 the Soviet cosmonaut Valentina Tereshkova became the first woman to go into space; there would be no American woman in orbit until Sally Ride flew aboard the space shuttle Challenger 20 years later.

Wally Funk is not one, however, to be deterred by long odds. Her triumph over Lovelace’s testing regime and her continuing determination to climb aboard a rocket ship are the least of her achievements. To date, she has more than 18,000 hours of flying under her belt; in 1971 she became the first woman successfully to complete the Federal Aviation Authority (FAA) General Aviation Operations Inspector Academy course; she was the first woman promoted to specialist in the FAA Systems Worthiness Analysis Program and the first woman to serve as air safety investigator with the National Transportation Safety Board in Washington, DC. A crash investigator’s job, she says, can be tough but the aim is simple: “You pursue an end result that hopefully will prevent another crash and save other lives.”

So she tells Sue Nelson, a British science journalist and broadcaster whose 2016 radio documentary, Women with the Right Stuff, was narrated by Wally. The least interesting parts of this book are concerned with Nelson’s attempts to turn the Tigger-ish Wally into a sober radio presenter. The rest is an odyssey of a friendship, as Nelson follows Wally’s dream to get into space before it’s too late.

The book takes them along the standard tourist trail of American space flight: Houston, Cape Canaveral – and over to the sites of the European Space Agency. Wally has spent $200,000 on a ticket for a flight on Virgin Galactic’s SpaceShipTwo, currently residing in a hangar in California’s Mojave Desert. Virgin’s commercial space-flight programme had a serious setback four years ago on its fourth powered flight when the VSS Enterprise suffered a catastrophic in-flight break-up; Michael Alsbury, the co-pilot, was killed.

Wally is undeterred. There is something almost unbearably poignant about her visit to Virgin’s HQ in London – anticlimactically near Paddington Station. The place is known as the “Battleship”, but Nelson describes it as a place David Brent would recognise. “We found staff, none of whom looked over thirty, working at computers in an open-plan area surrounding a kitchen-cum-bar area.” Sure enough there’s a shelf stacked with Branson’s books. “One of the Virgin Galactic team arrived and, surveying her workspace, asked, ‘What do you think? Cool, isn’t it?’”

Wally Funk’s Race for Space is a travelogue rather than a biography; Funk deserves the latter, and I could have done with more of Wally’s life and less of her failing to fasten her seatbelt in Sue’s rental car. Other books have been published about the Mercury 13, and earlier this year there was a fine Netflix documentary directed by David Sington and Heather Walsh. But Wally’s character shines through in Nelson’s book; not least her unfailing optimism. She’s not bitter, she says, that the Mercury 13 candidates weren’t taken more seriously. “I’m not that kind of person,” she says. “Why would I feel bitter? I’ve had nothing but great things happen.”

That’s true enough. But Wally isn’t finished fighting to fly; and as the 50th anniversary of the Moon landings approach, it’s sobering to have a reminder of the astronauts who might have been.

Erica Wagner is the author of “Chief Engineer: The Man Who Built the Brooklyn Bridge” (Bloomsbury)